where we have made use of the crossing relation
d∆ˆσgq→qX(xaPA, xbPB, PC/zc) =d∆ˆσqg→qX(xbPB, xaPA, PC/zc). (4.19) In terms of Mandelstams this exchange of pa and pb amounts to a replacement ofv, w by v0, w0 as given in Eq. (4.12). A similar bookkeeping as in the case ofqg→qX is necessary for all processes involving different incoming partons, whereas channels symmetric in the initial states as, e.g., gg → qX, must not be double-counted. The numerical results we obtain after carefully sorting out and normalizing all contributions to the individual subprocesses are then added up to the pionproduction cross sections at the NLO of pQCD.
4.2 First Numerical Results and Discussion
Let us now turn to a phenomenological study of single-inclusive pionproduction in hadronic collisions. We organize our calculation such that it resembles the experimental conditions of the most recent measurements at RHIC as closely as possible. We choose a c.m.s. energy of √
S = 200 GeV and consider pions which are scattered at an angle θ relative to one of the incoming proton beams with transverse momenta pT = |P~π|sinθ in the range 2≤pT ≤13 GeV. The pion’s scattering angle is parametrized by its pseudorapidity
η=−ln tan µθ
2
¶
. (4.20)
η is particularly suitable for the description of high-energy scattering reactions since it is additive under Lorentz boosts and can therefore easily be adjusted to different kinematical situations and frames. If not specified otherwise, we will integrate over the range of pseudorapidities relevant for the Phenix experiment at RHIC, |η| ≤ 0.38. It is useful to express the (polarized) differential cross section in terms ofpT and η,
Eπd3(∆)σ
d3Pπ = d3(∆)σ
dφ dη pTdpT, (4.21)
where the dependence on the azimuth is trivial. In the following, we will integrate over φ and η and consider only d(∆)σ/dpT.
In order to test the importance of NLO corrections we will present results computed both at LO and NLO. All (N)LO calculations are performed using (N)LO parton distri-bution and fragmentation functions and takingαs at (two) one loop(s). Unpolarized cross sections are evaluated with the CTEQ5 parton distributions [74], which include both a LO and an NLO parametrization. The spin structure of polarized protons is accounted for by the LO and NLO GRSV parton distributions [17]. Mostly, we will stick to their
“standard” scenario with a moderately positive gluon polarization. To study the sensi-tivity of the relevant double spin asymmetries to ∆g this choice will be contrasted to calculations with the “maximal gluon” scenario, which uses the maximal gluonic input at the starting pointµ0 of the evolution compatible with the positivity bound Eq. (2.47), i.e.,
∆g(x, µ0) = g(x, µ0). If not stated otherwise we will apply the pion fragmentation func-tions proposed by Kramer, Kniehl, and P¨otter [82], providing LO and NLO sets for neutral
0 1 2
0 5 10 15
dσNLO/dσLO
d∆σNLO/d∆σLO
pT [GeV]
d(∆)σ / dpT [pb / GeV]
unpolarized
polarized
NLO LO
102 104 106 108
0 5 10 15
Figure 4.2: Unpolarized and polarized differential cross sections at LO (dashed) and NLO (solid) for the reactionpp→π0X at √
S= 200 GeV. The lower panel shows the ratios of NLO and LO contributions.
and charged pions. This choice is supported by recent measurements of Phenix[100], as will be discussed below. If the values ofαsdiffer for the parton distributions and fragmen-tation functions used, we will calculate cross sections with the strong coupling associated with the evolution of the parton densities.
In Fig. 4.2 our thereby obtained results are presented for the polarized and unpolarized differential cross sections d(∆)σ/dpT at LO and NLO. As stated before, the unphysical scalesµr,µf, andµ0f are only an artifact of a perturbative calculation and thus arbitrary.
Here, for simplicity we have chosen all scales to coincide with the hard scale in the process, pT = µr = µf = µ0f. The lower panel of the figure displays the so-called “K-factor”, defined as the ratio of NLO to LO contributions,
K= d(∆)σN LO
d(∆)σLO . (4.22)
The K-factor for the unpolarized cross sections is almost constant over the pT-range considered. It increases only towards low values of the transverse momentum, where perturbative QCD is not supposed to be applicable. The functional behavior of K in
4.2 First Numerical Results and Discussion 63
LO (× 0.1)
NLO
pT [GeV]
d ∆σ / dp
T[pb / GeV]
1 10 102 103 104 105 106
0 5 10 15
Figure 4.3: Scale dependence of the polarized cross section forp~~p→π0X at LO and NLO. All scales are varied in the range pT/2 ≤µr =µf =µ0f ≤2pT. Solid lines correspond to the choice where all scales are set to pT. The LO results have been rescaled by a factor 0.1 for a better legibility.
the polarized case with generally smaller NLO corrections can be traced back to large cancelations and zeroes of the corresponding cross sections. At largerpT theK-factors for unpolarized and polarized cross sections approach each other. K is sometimes referred to as a “measure” for the importance of higher order corrections to hadronic processes. This proposition, however, has to be taken with some care. The K-factor depends strongly on the LO cross sections, which suffer from large scale uncertainties. In addition, new production mechanisms appearing for the first time at the NLO may lead to sizeable K-factors. However, similarly large corrections are not expected at higher orders.
A more reliable estimate on the impact of NLO corrections is provided by the improve-ment of scale dependence when extending a calculation from LO to NLO. We study this feature by a variation of the scales emerging in the computation, since setting all scales equal to pT is a suggestive, but by no means mandatory choice. Figure 4.3 presents the results obtained for the polarized cross sections at LO and NLO when varying all scales simultaneously in the range pT/2≤µr =µf =µ0f ≤2pT. Including NLO corrections to the cross sections reduces the scale uncertainty significantly, as expected from Eq. (2.12).
Whereas the LO results exhibit a strong scale dependence over the whole pT-range con-sidered, the NLO predictions are by far better constrained and vary only slightly as the scales are modified. As discussed in Sec. 2.1, this feature is one of the main motivations for performing a QCD calculation at NLO accuracy, since scale uncertainties eventually translate into theoretical errors for the extraction of ∆g. The inclusion of NLO corrections
leads to a reduction in theoretical uncertainties which are not sufficiently under control at the LO.
An excellent check for the reliability of our results is provided by a completely indepen-dent calculation of NLO corrections to single-inclusive pion production inpp-collisions in a Monte-Carlo approach by de Florian [106]. Rather than analytically phase space integrat-ing all matrix elements in n dimensions as we did, a numerical Monte-Carlo integration has been performed in [106]. This technique makes use of the singularity structure of the 2→3 matrix elements, which allows for an explicit separation and cancelation of all poles by hand [93, 107]. The finite remainders are numerically phase space integrated in four dimensions, which is quite time-consuming and tedious, making the technique inappro-priate for extracting parton densities in a global analysis of data, which will be discussed in Sec. 4.3. However, the Monte-Carlo approach has the advantage of a higher flexibility than our largely analytical method. Experimental cuts can easily be implemented, and an extension of the code to different observables such as jet production is relatively straight-forward. The outcome of a comparison of our results to the ones of de Florian, however, was that both methods give the same answer.
Having ensured the reliability of our calculation in various ways, we are in a position to compare our results to experiment. In the following we set all scales to pT by default.
Figure 4.4 shows our predictions for the unpolarized π0-production cross section at NLO together with recent data from the Phenix collaboration [100]. To get an estimate on the impact of different fragmentation functions we have performed the calculation with the KKP set as well as with Kretzer’s parametrization [83]. The agreement of the data points with our theoretical prediction is encouraging, as it gives some confidence in the applicability of perturbation theory, down to rather low values ofpT '1.5 GeV. In partic-ular, our results obtained with the KKP parametrization agree well with the measurement over almost the whole pT-range considered. The calculation performed with Kretzer’s set of fragmentation functions lies significantly below the data. This behavior can be traced back to a much smaller Dgπ in the latter set, which is not constrained by the e+e− data used for the extraction of the fragmentation functions [82, 83], as discussed in Sec. 2.5.
Due to the dominance of gluonic channels at low-to-intermediate pT, in this range the effects of a smallerDgπ are particularly pronounced. Since the KKP parametrization gives a good description we will perform our further analyses using this set for describing pion fragmentation.
In Fig. 4.5 data from the Star collaboration [101] are compared to our NLO calcu-lation. Currently, the acceptance of the Star detector is such that pions can only be detected at very small scattering angles, corresponding to rapidities up to η'4. There-fore, it allows, in principle, to access the parton distributions at smaller values ofx than, e.g., the Phenix experiment, where only events at central rapidity are detected, as can be seen from the definition of xa and xb in Eq. (4.6). Our calculation is performed at the RHIC c.m.s. energy of √
S = 200 GeV and fixed η = 3.8. The results are displayed in terms of the pion’s energy,Eπ =pTcoshη, rather than its transverse momentum. Our pre-dictions obtained with the KKP fragmentation functions [82] agree again rather well with the data over the whole Eπ range considered. As before, using Kretzer’s parametrization yields results lying somewhat below the data.
4.2 First Numerical Results and Discussion 65
)
3c ⋅
-2GeV ⋅ (mb
3/dp σ
3E*d
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
1
a)
PHENIX Data KKP FF Kretzer FF
(%) σ / σ∆
-40 -20 0 20
40
b)
0 2
4
c)
(GeV/c) pT
0 5 10 15
0 2
4
d)
(Data-QCD)/QCD
Figure 4.4: Unpolarized cross sectionEπ
¡d3σ/d3Pπ
¢forpp→π0X at √
S= 200 GeV: a) Com-parison of our NLO results obtained with the fragmentation functions of KKP [82] (solid line) and Kretzer [83] (dotted line) to the data from Phenix [100]; b) Relative statistical errors (points) and systematic errors (bands) of data; c, d) Relative differences between data and the theoretical prediction with KKP and Kretzer, respectively. The figure has been taken from Ref. [100].
10 -1 1 10
25 30 35 40 45 50 55 60 65
NLO pQCD calc.
Data 3.4<η<4.0
π0 mesons (〈η〉=3.8)
Eπ (GeV) E d3 σ/dp3 (µb c3 /GeV2 )
KKP F.F. (η=3.8) Kretzer F.F. (η=3.8)
Normalization Uncertainty = 17%
〈pT〉 = 1.4 1.6 1.8 2.1 2.2 GeV/c
Figure 4.5: Data for the unpolarized cross sectionEπ
¡d3σ/d3Pπ
¢forpp→π0Xat√
S= 200 GeV from the Star collaboration [101], compared to our NLO results using the KKP fragmentation functions [82]. The figure is taken from Ref. [101].